4 - Typical Finite Elements

Summary

The finite-element method is a powerful tool to stimulate the creativity of engineers, scientists, and applied mathematicians. Therefore it is not surprising that many elements have been presented in the literature and are available for a range of applications.

A finite element is defined not only by a geometric subdomain, or even the explicit mathematical expression of the shape functions. A finite element also includes the specification of whether it is an isoparametric element (as we shall essentially consider here) and the definition of the space integration formulas. In a broader sense we can also consider that the space discretization procedure for each variable, the time integration procedure when required, and the mathematical formulation of the physical problem also contribute to the FE definition.

Evaluation of the behavior of an element type will depend on the kind of problem that is considered, the required accuracy, and the time and effort allocated toward code development. In the realm of metal forming, the subject of this book, the choice of element formulation depends primarily on the following considerations.

1. Mode of deformation: the chosen element must approximate real material stiffness in all modes of deformation important to the physical problem being simulated. An element can be too stiff in some modes, producing “locking” for example, particularly when incompressibility constraints are considered; or it can be too compliant, producing “hourglass” and other spurious low-energy deformation modes.